A Few Notes about Notations

Several days ago I attached a page to this blog tentatively titled "Spiral Set." I say "tentatively" because it's simply a working title referring to the post "The Form" that began this thread on deep connections. The title will likely change to something more accurately descriptive when I consider this set of eight slides is correct and complete (or when someone tells me that this "super-group"/"hyper-group"/? has been named and studied elsewhere, which would save me a lot of work). I decided to publish this nearly complete set now since nomenclature starts to get cumbersome in describing and applying these inter-related permutations in traditionally marginally- or un-related fields. So a word or two here about the eight slides.

Each slide contains four related permutations. Talking through just one of the slides will explain the other seven. I'm using permutations of order 6 (remember the sestina) to illustrate, but the basic set can be of just about any size. I've chosen the "COIL" slide as an example because I'll be moving soon from the "perfect (Faro) shuffle" in cards to its application in computers.

Some readers will already have noticed that at this point I have introduced four parallel notations, i.e., four different ways of describing (or, better, seeing) the same result.

(1) Open continuous paths. At the top of the slide is a diagram [appearing only in the first four slides as of now – I haven't included the other four yet] showing a 2-dimensional continuous path that crosses once through each of the six points on the line (imagine each point labelled consecutively 1 through 6). Begin at one of the "loose ends" of the path, two of which are "outside" ends and two of which are "inside." Thus outL means start following the path at the outside end on the left resulting in the path on the line [142536] ; outR means start at the outside end on the right, [635241]; inR starts the path at the inside right loose end, [415263]; inL begins at the inside left end, [362514].

(2) Comb combinations. All of these permutations (on all eight slides) are based on dividing the basic set in half in one of two ways: Left and Right halves, L={123} R={456}, and Odd and Even halves, O={135} E={246}.[1]  Each of these subsets is then expressed as an ordered set according to whether the integers increase (→: 123, 456, 135, 246) or decrease (←: 321, 654, 531, 642). Finally, the oriented subsets are either INTERLeaved as in this example or CONCATenated (see slide 1, for example) beginning with whichever subset is written first. Thus, INTERL(L→R→) means write 1 2 3 and then, starting after the 1, interleave 4 5 6 yielding the permutation [142536].[2]  CONCAT(L→R→) = [123456] (see slide 6). INTERL(L←R←) = [362514] (slide 2 above). CONCAT(L←R←) = [321654] (slide 6). INTERL(E←O→) = [614325] (slide 7). Etc. for a total of 32 comb combinations. Also note that the four comb combinations on each of the eight slides are related by both a FLIP operation (O↔E or L↔R, the red lines) and a REVerse orientation operation (← ↔ →, the green lines).

The comb notation and all of the operations it suggests might seem at first to be a highly convoluted way of distinguishing this bunch of 32 permutations (technically only 30 because two of them [123456] and [654321] are repeated) out of all possible permutations. But it's helpful to remember that the comb idea generating all these permutations originated with the medieval definition of the sestina form as retrogradatio cruciata. Once you have the acorn, sooner or later the entire oak will appear for you. But there are two more notations left – the more traditional ones from mathematics.

(3) 1-to-1 map (Cauchy two-line notation). This is the simplest way of representing any permutation and is generally used when introducing the concept in elementary math courses.

It means no more than the top row "goes to" the bottom row: 1→1, 2→4, 3→2, etc. This is often abbreviated in "one-line notation" where the consecutive integers in the top row are assumed; all that's needed is to state the outcome, the bottom row (see the double-line enclosed boxes in slide 2 above).

(4) Closed continuous paths (cyclic notation). This is analogous to the open continuous path in that it cycles through the elements in the string being permuted. Whereas the "open path"notation described in (1) leads through the initial string from one end to another (e.g., 1→4→2→5→3→6), occasionally revealing some nice symmetries, cyclic notation creates one or more closed loops. Referring to the previous example in two-line notation, first note that 1 and 6 are "fixed points" in the example – they don't change in the permutation. Next, go to 2 in the top row and follow the path through: 2→4,4→5,5→3,3→2. tracing the closed path 2→4→5→3→(2). In the traditional notation for cyclic notation this would be written: (1)(6)(2453) as it is on slide 2 above. While I have never seen it presented this way, we could present cyclic notation visually in much the same way the open path notation was presented:

Using cyclic notation it becomes easy to determine any permutation's inverse, e.g., having gone from [123456] to [142536], what permutation will immediately take me back to [123456]? The answer is to simply read each cycle backward. So (1) and (6) are unchanged, but (2453) becomes (3542) which, remembering this is cyclic, is the same as (2354) or (4235) or (5423).  But notice that the inverse we are looking for, (1)(6)(3542) is not available on slide 2. It can be found on slide 1. On inspection we find that slides 1 and 2 are inverse-related, as well as the pair of slides 3 and 4. For the rest, however, the inverse of any permutation is found on the same slide.

A summary of the relationships between the eight slides can be found on the page "Spiral Set Stack."

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[1] If the order of the basic set is odd, say 7, all of these relationships will still work and create analogous structures. To work with an odd order, begin by extending the open path diagrams in order to get around the difficulty of dividing the basic set exactly in half. It's an interesting exercise which I'll leave to the reader.

[2] Sorry. Typography problem here. Think of the arrows to the right of a letter within the text as being above the letter as in the slide.

Jeu de Cartes

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged.

 ––Poincaré

The idea that someone with a bit of natural dexterity and a lot of practice can trace any card's path as it travels through several riffle shuffles of a deck is amazing to most of us (and a good reason we amateurs should stay away from Blackjack tables and Poker games). The above video by Kevin Houston gives an excellent demonstration of how card tracking is possible using a "perfect shuffle" and then goes on to explain the math behind why this particular shuffle works as it does. In passing, musicians will recognize one of the techniques since it's based on modular arithmetic––the same elementary principle as the one at work behind the "octave equivalence" and "pitch class" concepts. But what we're interested in here is still mostly the "geometries of permutations."

Another interesting card shuffle is often associated with the mathematician Gaspard Monge (1746-1818), a colorful character active during and after the French Revolution and usually thought of as the inventor of descriptive geometry. What is often called the "Mongean shuffle" works like this. . . .

We'll select only six cards from the deck to make the principle clear, keeping in mind that, once understood, it can be applied to the entire deck of 52 cards. Let's say we have, arranged from top to bottom, the Ace, 2, 3, 4, 5, and 6 of clubs. Now, ignoring the mechanics of how this shuffle might be performed in practice by a dealer or magician (it's rather clumsy compared to the riffle illustrated in the video), we lay out a single Mongean shuffle of these six cards face up. Begin by laying down the top card, the Ace, then place the next card, a deuce, on the table to the left of the Ace, then the three to the right of the Ace, the four to the left of the deuce, the five to the right, and finally the six to the left. The result is the re-ordering 6, 4, 2, Ace, 3, 5:

Flipping this map diagonally and repeating the procedure, we can follow any card through six shuffles until it returns to its original position.

Why does this map look familiar? (As if you haven't guessed already.) But "looks familiar" isn't enough. Let's try applying a procedure that, once again, will be familiar to most musicians – or at least to those who have played/looked at Bach's Musical Offering.[1]

Here again is the basic Mongean shuffle:

Let's get the cards out of the way to follow the moves more easily. This gives us a sort of placeless map .... (sorry) ...

Start by flipping this map over horizontally, resulting in a retrograde map.

Then flip that result over vertically, resulting in an inversion of the retrograde.

Then lay the cards back on the table using the new map to see what this retrograde-inversion shuffle results in.

The retrograde-inversion of the Mongean shuffle produces precisely the same map we encountered in the sestina.

But we're not finished quite yet. Trying to visualize the Mongean shuffle as a transformed version of a spiral (the way we initially described the sestina) is a bit of a stretch. Still, following a curvy path, it almost makes sense to see how the spiral:

might be seen as a tangled/untangled version of Monge, which itself can be seen to resemble a snake twisting back on itself or the action of a wave rebounding from the side of a pool:

If we want to get away from continuous "curving actions" but still stay with some sort of helpful visualization that gives the same results, we might think of these two procedures as things to do with two three-pronged combs. Making one comb out of the left side and one out of the right, we reverse one of them, stretch them both a bit, and then interleave the result. Then the spiraling looks like this in the corresponding comb version:

Starting back at the consecutive numbers once more, this time dividing the positions into odd and even, we separate them again, turn one of them backward again, and then recombine by concatenation.  So the wave in its corresponding comb version looks like this:

*

Ideomorphically, of course, the two worlds can't be mixed––playing cards are not words and card shuffling is not poetry. And neither, again ideomorphically, is a snake or a spiral a card trick or a poem. And that's just the point, because we are searching for sub-surface isomorphic actions–beyond metaphor and analogy–that connect otherwise unrelatable worlds. In the isomorphic underworld, not only can a hair comb be a metaphorically useful object-symbol within a poem, it can also provide a translucent structural basis for the poem. In fact, the first of the two "comb moves" above is precisely the way the sestina was often described, rather than the spiral description I chose to begin with. The technique in medieval times was known as retrogradatio cruciata–"backward crossing."[2]

Next: Just what is multi-tasking parallel processing, anyway? ––––>

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[1] I mention Bach in this passing reference rather than Schoenberg, whom many would prefer to focus on as the more obvious culprit. I did this to keep my background emphasis that all these tricks are ancient & didn't just spring into being starting in the early 20th century. It's relatively easy to identify the inventor of the steam engine, but not so easy to identify the inventor of the wheel. By the time JS Bach came on the scene, many of these tricks/transformations had already long been in use. For a fun time with the Bach canons in the Musical Offering, there are two interesting web sites – one in a math version and one in a non-math version.

[2] Treated as religious symbolism, this term might also suggest anything from dark word play to heresy to outright satanism. But I have never read any suggestion of this elsewhere, so this must remain my personal fantasy. (Still, an interesting potential plot twist for a novelist or playwright, though?)